Was reading this again and was wondering if you have tried silicones?
Sound like your urethanes have too much traction. In my experience both on a DC704A car I had a few years back and my current DC706 car you don't need much traction with so much weight directly on the axle.
The old tires (which are still soft and pliable but not nearly as grippy as modern urethanes) on my current car work fine and just have some tilting that is still controllable. I plan to radius the edges at some point which I think will make it even better.
The DC704A car had a Fred's Enterprise chassis and had similar problems that you have. Tried a pair of silicones and they worked well. Hooked up good but had slide in the corners with a short, smooth drift.
I would just try it out and see if it work for you. If you don't have some that will work, I have a pair that might. Possibly.